Transcript Two routes to the one-dimensional discrete nonpolynomial

Alternative routes to the 1D discrete
nonpolynomial Schrödinger equation
Boris Malomed
Dept. of Physical Electronics, Faculty of Engineering, Tel Aviv
University, Tel Aviv, Israel
Goran Gligorić and Ljupco Hadžievski
Vinća Institute of Nuclear Sciences, Belgrade, Serbia
Aleksandra Maluckov
Faculty of Sciences and Mathematics, University of Niš, Niš, Serbia
Luca Salasnich
Department of Physics “Galileo Galilei”
Universitá di Padova, Padua, Italy
The topic of the work: approximation of the GrossPitaevskii equation for BEC by means of discrete models,
in the presence of a deep optical lattice. We report
derivation of a new model, and its comparison with the
previously know one. A simulating discussion with Dmitry
Pelinovsky is appreciated.
Some revious works on the topic of the discrete approximation for BEC:
A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001);
F.Kh. Abdullaev, B. B. Baizakov, S.A. Darmanyan, V.V. Konotop & M.
Salerno, Phys. Rev. A 64, 043606 (2001);
G.L. Alfimov, P. G. Kevrekidis, V.V. Konotop & M. Salerno, Phys. Rev. E
66, 046608 (2002);
R. Carretero-González & K. Promislow, Phys. Rev. A 66, 033610
(2002);
N.K. Efremidis & D.N. Christodoulides, ibid. 67, 063608 (2003);
M. A. Porter, R. Carretero-González, P. G. Kevrekidis & B. A. Malomed,
Chaos 15, 015115 (2005);
G. Gligorić, A. Maluckov, Lj. Hadžievski & B.A. Malomed, ibid. 78,
063615 (2008).
The starting point: the rescaled threedimensional (3D) Gross-Pitaevskii equation
(GPE) for the normalized mean-field wave
function. The GPE includes the transverse
parabolic trapping potential and the longitudinal
(axial) optical-lattice (OL) periodic potential:
  1 2
1 2
i
      V0 cos(2qz )   x  y 2   2  | 
t  2
2
where   2 Nas /a , a 

|  ,

2
/  m  ,  is the trapping
frequency, N is the number of atoms, and as is the
scattering length. The repulisve/attractive interactions
between atoms correspond to   0/  0, respectively.
The first (“traditional”) way of the derivation of the 1D
discrete model: start by reducing the dimension of the
continual equation from 3 to 1, and then discretize the
effective 1D equation.
The reduction of the dimension from 3 to 1 is carried
out through the following ansatz:
 x2  y 2 
  x, y , z , t  
exp   2
f ( z , t ),

  z , t 
 2 ( z , t ) 
1
with normalization



| f ( z , t ) |2 dz  1.
The resulting nonpolynomial Schroedinger equation (NPSE)
[L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 (2002)]:
f  1  2
1  (3 / 2) | f |2 
i
 
 V0 cos  2qz  
 f.
2
2
t  2 z
1   | f | 
This one  dimensional equation admits the description of the
collapse if  < 0, at the critical density : (| f | 2) cr  1/ .
The discretization of the one-dimensional NPSE leads to
the DNPSE [A. Maluckov, Lj. Hadžievski, B. A. Malomed,
and L. Salasnich, Phys. Rev. A 78, 013616 (2008)], through
the expansion over a set of localized wave functions
(e.g., Wannier functions):
f ( z , t )   f n (t )Wn ( z ).
n
The result:
 
df n  1 
1
2
i
 
 1  g | f n |     f n  C  f n 1  f n 1 
2
 
dt  2  1  g | f n |
 
 
g | f n |2

fn .
2
1  g | fn |
This discrete equation also admits the possibility of the
collapse for g  0, at the critical value of the on-site
density, | f n |2  1/ g .
The new (“second”) route to the derivation of the 1D
discrete model: first perform the discretization of the
underlying 3D GPE in the direction of z, and then perform
the reduction of the dimension. The two operations are
expected to be non-communitative.
Assume the expansion of the 3D wave function:
 ( x, y, z , t )   n ( x, y, t )Wn ( z ).
n
This leads to the semi - discrete equation,
n  1 2 1 2

2
i
       x  y     n
t  2
2

C n 1  n 1   2 g | n |2 n .
The reduction of the semi-discrete
equation to the 1D discrete form.
The dimension-reducing ansatz:
 x2  y 2 
n ( x , y , t ) 
exp  
 f n ( z , t ).
2
 n ( z , t )
 2 n ( z , t ) 
The substitution of the ansatz into the semi-discrete
1
equations and averaging in the plane of  x, y  leads
the following system of coupled equations for the
complex 1D discrete time-dependent wave function,
f n (t ), and the real transverse width,  n (t), that may
also be time-dependent functions.
At the first glance, this system seems very complex, and
completely different from the DNPSE equation for the
discrete wave function, fn(t), which was derived by means
of the “traditional approach”. Our objective is to find and
compare families of fundamental discrete solitons in
both models, the “traditional simple”, and “new
complex” ones.
The distinction between the “traditional” and “new” models
is also seen in the difference between their Hamiltonians,
while they share the same expression for the norm (P):
P   | f n |2 ,
n


H old   C | f n  f n 1 |  1  g | f n | | f n | ,
n
2
2
2
1  1

 n n 1
2
H new   f n   2   n     f n  2C 2
2
 n   n 1
n
 
2 n
g
*
*
4
  f n f n 1  f n 1 f n  
|
f
|
.
n
2
2 n
{
*
}
First, we consider the modulational instability of CW
(uniform) waves, fn(t) = U exp(-iμt), in the case of the
attractive nonlinearity, g < 0. The CW solutions are
unstable in regions below the respective curves (q is the
wavenumber of the modulational perturbations):
Comparison of the shapes of unstaggered onsitecentered discrete solitons: transverse widths (σn) at the
central site of the soliton, and two sites adjacent to the
center, in the two models, for C = 0.2 and C = 0.8. Chains
of symbols: the “old” model; curves: the “new” one.
The same comparison for unstaggered intersitecentered solitons:
Stationary soliton solutions with chemical
potential μ are looked for as fn = Fn exp(-iμt),
with real Fn, for two soliton families: onsitecentered and intersite-centered ones. Then, the
families may be described by dependences of the
norm vs. the chemical potential,
P(  )   Fn (  ).
2
n
The comparison of the P(μ) characteristics for the
family of onsite-centered solitons, at C = 0.2 and
C = 0.8. Note that the VK stability criterion,
dP/dμ < 0, generally, does not apply to the model
with the nonpolynomial nonlinearity.
The same comparison for families of intersitecentered solitons, also for C = 0.2 and C = 0.8:
A similar comparison of values of the free energy for both
models, G = H – μP, again for C = 0.2 and C = 0.8. Left
and right curves in both panels pertain to on-site and
inter-site solitons, respectively:
The comparison of dynamical properties of the discrete solitons in
both models. In particular, real eigenvalues (“ev”), accounting for the
instability of a part of the family of the onsite-centered solitons (a)
and of the intersite-centered soliton family (b), at C = 0.8, as
functions of μ, have the following form in both models (the intersite
solitons are completely unstable in both models; onsite solitons have
their stability region):
CONCLUSIONS
There are two alternative ways to derive the 1D
discrete equation for BEC trapped in the
combination of the tight transverse confining
potential and deep axial optical lattice:
(1) First, reduce the dimension from 3 to 1,
arriving at the nonpolynomial NLS equation,
and then subject it to the discretization; or
(2) First, apply the discretization to the 3D
equation, reducing it to a semi-discrete form,
and then eliminate (by averaging) the two
transverse dimensions.
Because the reduction of the dimension
and the discretization do not commute,
these alternative routes of the derivation
lead to two models which seem
completely different: the earlier known
discrete nonpolynomial NLS equation
(obtained by means of the former method),
or the model obtained by means of the
latter method, which, apparently, has a
much more complex form.
However, despite the very different form of the two
models, they produce nearly identical results
for the shape of unstaggered discrete solitons of
both the onsite- and intersite-centered types,
and virtually identical conclusions about their
stability.
Additional numerical analysis demonstrates that
the character of the collapse in both discrete
models is also essentially the same (note that
both models are capable to predict the collapse
in the framework of the 1D discrete
approximation), as well as regions of the
mobility of the solitons.
The general conclusion:
The approximation of the dynamical
behavior of the 3D condensate in the
present setting by the 1D discrete model is
reliable, as two very different models
yield almost identical eventual results.